Question: Given $ m \angle QPR = 7x - 54$, and $ m \angle RPS = 7x + 80$, find $m\angle RPS$. $P$ $Q$ $S$ $R$
Solution: From the diagram, we see that together ${\angle QPR}$ and ${\angle RPS}$ form ${\angle QPS}$ , so $ {m\angle QPR} + {m\angle RPS} = {m\angle QPS}$ Since $\angle QPS$ is a straight angle, we know ${m\angle QPS = 180}$ Substitute in the expressions that were given for each measure: $ {7x - 54} + {7x + 80} = {180}$ Combine like terms: $ 14x + 26 = 180$ Subtract $26$ from both sides: $ 14x = 154$ Divide both sides by $14$ to find $x$ $ x = 11$ Substitute $11$ for $x$ in the expression that was given for $m\angle RPS$ $ m\angle RPS = 7({11}) + 80$ Simplify: $ {m\angle RPS = 77 + 80}$ So ${m\angle RPS = 157}$.